By Loring W. Tu

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Wedge product and scalars Let V be a vector space. For a, b ∈ R, f ∈ Ak (V ), and g ∈ Aℓ (V ), show that a f ∧bg = (ab) f ∧g. 7. Transformation rule for a wedge product of covectors Suppose two sets of covectors on a vector space V , β 1 , . . , β k and γ 1 , . . , γ k , are related by βi = k ∑ aij γ j , i = 1, . . , k, j=1 for a k × k matrix A = [aij ]. Show that β 1 ∧ · · · ∧ β k = (det A) γ 1 ∧ · · · ∧ γ k . 8. Transformation rule for k-covectors Let f be a k-covector on a vector space V .

Show that α 1 ∧ · · · ∧ α k = 0 if and only if α 1 , . . , α k are linearly independent in the dual space V ∨ . * Exterior multiplication Let α be a nonzero 1-covector and γ a k-covector on a finite-dimensional vector space V . Show that α ∧ γ = 0 if and only if γ = α ∧ β for some (k − 1)-covector β on V . 34 §4 Differential Forms on Rn §4 Differential Forms on Rn Just as a vector field assigns a tangent vector to each point of an open subset U of Rn , so dually a differential k-form assigns a k-covector on the tangent space to each point of U.

Propositions A and B express the property d 2 = 0 of the exterior derivative on open subsets of R3 ; these are easy computations. Proposition C expresses the fact that a 1-form on R3 is exact if and only if it is closed. Proposition C need not be true on a region other than R3 , as the following well-known example from calculus shows. Example. If U = R3 − {z-axis}, and F is the vector field −y x , ,0 x2 + y2 x2 + y2 F= on R3 , then curlF = 0, but F is not the gradient of any C∞ function on U. The reason is that if F were the gradient of a C∞ function f on U, then by the fundamental theorem for line integrals, the line integral C − y x2 + y2 dx + x x2 + y2 dy over any closed curve C would be zero.